The path of a charged particle in a uniform magnetic field, when the velocity and the magnetic field are perpendicular to each other is a

Options

A. circle
B. parabola
C. helix
D. straight line

Answer

To solve the question regarding the path of a charged particle in a uniform magnetic field when its velocity and the magnetic field are perpendicular to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- We have a charged particle with charge \( q \) moving with a velocity \( \vec{v} \) in a uniform magnetic field \( \vec{B} \).
- The velocity \( \vec{v} \) is perpendicular to the magnetic field \( \vec{B} \).

2. **Applying the Lorentz Force**:
- The force acting on the charged particle due to the magnetic field is given by the Lorentz force equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
- Since \( \vec{v} \) and \( \vec{B} \) are perpendicular, the magnitude of the force can be simplified to:
\[ F = qvB \]
- This force acts as the centripetal force that keeps the particle in circular motion.

3. **Centripetal Force and Circular Motion**:
- For an object moving in a circle of radius \( r \) with speed \( v \), the centripetal force is given by:
\[ F = \frac{mv^2}{r} \]
- Setting the magnetic force equal to the centripetal force, we get:
\[ qvB = \frac{mv^2}{r} \]

4. **Finding the Radius of the Circular Path**:
- Rearranging the equation to solve for \( r \):
\[ r = \frac{mv}{qB} \]
- This shows that the radius of the circular path depends on the mass \( m \) of the particle, its velocity \( v \), the charge \( q \), and the magnetic field strength \( B \).

5. **Conclusion**:
- Since the charged particle experiences a constant magnetic force that is always perpendicular to its velocity, it will move in a circular path.
- Therefore, the path of the charged particle in a uniform magnetic field, when the velocity and the magnetic field are perpendicular to each other, is a **circular path**.

### Final Answer:
The path of a charged particle in a uniform magnetic field, when the velocity and the magnetic field are perpendicular to each other, is a **circular path**.

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